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Enunciado

Álgebra Linear e Geometria Analítica - Anton - Álgebra Linear- Ed: 10º - Capítulo 1.4 - Ex. 2c

Sejam A = ([[2,-,1,3],[0,4,5],[-,2,1,4]]) , B = ([[8,-,3,-,5],[0,1,2],[4,-,7,6]]) , C = ([[0,-,2,3],[1,7,4],[3,5,9]]) , a = 4 e b = - 7 , mostre que (B + C) A = B A + C A

Passo 1

Esse problema aborda, de maneira não generalizada, a demonstração de algumas propriedades algébricas. Nesse caso, vamos mostrar que (B + C) A = B A + C A . Para isso, vamos resolver os dois lados da equação e comparar para ver se são, de fato, iguais.

Assim, montemos o primeiro lado:

(([[8,-,3,-,5],[0,1,2],[4,-,7,6]]) + ([[0,-,2,3],[1,7,4],[3,5,9]])) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

E o segundo:

([[8,-,3,-,5],[0,1,2],[4,-,7,6]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]]) + ([[0,-,2,3],[1,7,4],[3,5,9]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

Passo 2

Resolvendo o primeiro lado, temos:

(([[8,-,3,-,5],[0,1,2],[4,-,7,6]]) + ([[0,-,2,3],[1,7,4],[3,5,9]])) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

([[8,+,0,-,3,+,(,-,2,),-,5,+,3],[0,+,1,1,+,7,2,+,4],[4,+,3,-,7,+,5,6,+,9]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

([[8,-,5,-,2],[1,8,6],[7,-,2,15]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

([[8,xx,2,+,(- 5),(0),+,(,-,2,),xx,(,-,2,),8,xx,(- 1),+,(- 5),xx,4,+,(,-,2,),xx,1,8,xx,3,+,(- 5),xx,5,+,(,-,2,),xx,4],[1,xx,2,+,8,xx,0,+,6,xx,(,-,2,),1,xx,(,-,1,),+,8,xx,4,+,6,xx,1,1,xx,3,+,8,xx,5,+,6,xx,4],[7,xx,2,+,(- 2),xx,(0),+,15,xx,(,-,2,),7,xx,(,-,1,),+,(- 2),xx,4,+,15,xx,1,7,xx,3,+,(- 2),xx,5,+,15,xx,4]])

([[16,-,0,+,4,-,8,-,20,-,2,24,-,25,-,8],[2,+,0,-,12,-,1,+,32,+,6,3,+,40,+,24],[14,-,0,-,30,-,7,-,8,+,15,21,-,10,+,60]])

([[20,-,30,-,9],[-,10,37,67],[-,16,0,71]])

Passo 3

Resolvendo o segundo lado da equação:

([[8,-,3,-,5],[0,1,2],[4,-,7,6]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]]) + ([[0,-,2,3],[1,7,4],[3,5,9]]) ([[2,-,1,3],[0,4,5],[-,2,1,4]])

O primeiro produto:

([[8,xx,2,+,(- 3),(0),+,(,-,5,),xx,(,-,2,),8,xx,(- 1),+,(- 3),xx,4,+,(,-,5,),xx,1,8,xx,3,+,(- 3),(5),+,(,-,5,),xx,(,-,4,)],[0,xx,2,+,1,xx,0,+,2,xx,(,-,2,),0,xx,(,-,1,),+,1,xx,4,+,2,xx,1,0,xx,3,+,1,xx,5,+,2,xx,(,-,4,)],[4,xx,2,+,(- 7),xx,(0),+,6,xx,(,-,2,),4,xx,(,-,1,),+,(- 7),xx,4,+,6,xx,1,4,xx,3,+,(- 7),xx,5,+,6,xx,(,-,4,)]])

([[16,+,0,+,1,0,-,8,-,1,2,-,5,24,-,15,+,20],[0,+,0,-,4,0,+,4,+,2,0,+,5,-,8],[8,+,0,-,12,-,4,-,28,+,6,12,-,35,+,2,4]])

([[26,-,25,29],[-,4,6,-,3],[-,4,-,26,1]])

O segundo produto:

([[0,xx,2,+,(- 2),xx,0,+,3,xx,(,-,2,),0,xx,(,-,1,),+,(- 2),xx,4,+,3,xx,1,0,xx,3,+,(- 2),xx,5,+,3,xx,4],[1,xx,2,+,7,xx,0,+,4,xx,(,-,2,),1,xx,(,-,1,),+,7,xx,4,+,4,xx,1,1,xx,3,+,7,xx,5,+,4,xx,4],[3,xx,2,+,5,xx,0,+,9,xx,(,-,2,),3,xx,(,-,1,),+,5,xx,4,+,9,xx,1,3,xx,3,+,5,xx,5,+,9,xx,4]])

([[0,+,0,-,6,0,-,8,+,3,0,-,10,+,12],[2,+,0,-,8,-,1,+,28,+,4,3,+,35,+,16],[6,+,0,-,18,-,3,+,20,+,9,9,+,25,+,36]])

([[-,6,-,5,2],[-,6,31,54],[-,12,26,70]])

Realizando a soma:

([[26,-,25,29],[-,4,6,-,3],[-,4,-,26,1]]) + ([[-,6,-,5,2],[-,6,31,54],[-,12,26,70]])

([[2,6,-,6,-,2,5,-,5,29,+,2],[-,4,-,6,6,+,31,-,3,+,54],[-,4,-,1,2,-,26,+,26,1,+,70]])

([[20,-,30,-,9],[-,10,37,67],[-,16,0,71]])

Resposta

Ambos os lados são iguais. Logo, está demonstrado que (B + C) A = B A + C A

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